Let's solve for the exact values of $\cos\left(\frac{\pi}{3}\right)$ and $\cos\left(-\frac{\pi}{3}\right)$.

1. $\cos\left(\frac{\pi}{3}\right)$

We know from the unit circle or from common trigonometric values that:

$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

2. $\cos\left(-\frac{\pi}{3}\right)$

Since the cosine function is an even function, it holds that:

$\cos(-x) = \cos(x)$

So,

$\cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

Final answers:

• $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$
• $\cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}$

Would you like any further details or have any questions on this? Here are some follow-up questions:

1. What is the difference between even and odd trigonometric functions?
2. How do we determine trigonometric values from the unit circle?
3. What are the exact values for $\sin\left(\frac{\pi}{3}\right)$ and $\tan\left(\frac{\pi}{3}\right)$?
4. How does the symmetry of the unit circle help us calculate trigonometric values for negative angles?
5. What is the relationship between radians and degrees in trigonometry?

Tip: Always remember that the cosine function is even, meaning $\cos(-x) = \cos(x)$, which can simplify many calculations.